Chapter 6 Section 6.1 – Classifying Quadrilaterals Objectives: To define and classify special types of quadrilaterals

Definitions -> Special Quadrilaterals Parallelogram -> a quadrilateral with both pairs of opposite sides parallel Rhombus -> a parallelogram with four congruent sides Rectangle -> a parallelogram with four right angles Square -> a parallelogram with four congruent sides and four right angles Kite -> a quadrilateral with two pairs of adjacent sides congruent and no opposite sides congruent Trapezoid -> a quadrilateral with exactly one pair of parallel sides Isosceles Trapezoid -> a trapezoid whose nonparallel opposite sides are congruent

Ex: Classifying a Quadrilateral Classify the following figures in as many ways as possible

Ex: Using the Properties of Special Quadrilaterals Find the values of the variable for the kite T 2y + 5 x + 6 J K 2x + 4 3x - 5 B

Ex: Find the values of the variables for the rhombus. Then find the lengths of each side. L 3b + 2 N 5a + 4 3a + 8 S T 4b - 2

Homework #31 Due Tuesday (December 04) Page 308 – 309 # 1 – 12 all # 13 – 25 all

Section 6.2 Properties of Parallelograms Objectives: To use relationships among sides and among angles of parallelograms To use relationships involving diagonals of parallelograms or transversals

Theorem 6.1 Opposite sides of a parallelogram are congruent Consecutive Angles -> angles of a polygon that share a side. Since a parallelogram has opposite sides parallel, consecutive angles are same-side interior angles (supplementary) D C <A and <B <B and <C <C and <D <D and <A A B

Ex: Using Consecutive Angles Find the measure of each missing angle in parallelogram RSTW S T 112° R W

Theorem 6.2 Ex: Opposite angles of a parallelogram are congruent Find the value of x and y in parallelogram PQRS. Then find the measures of each side and each angle. Q 3x - 15 R (3y + 37)° (6y + 4)° P S 2x + 3

Theorem 6.3 The diagonals of a parallelogram bisect each other JH congruent LH KH congruent MH J K H M L

Ex: Find the value of x and y. Then find the value of each line segment B C 2x x + 1 E 3y - 7 y A D

Theorem 6.4 If three (or more) parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal. A B C D E F

Homework #32 Due Wednesday (Dec 05) Page 315 – 316 # 1 – 13 all # 14 – 30 even

Section 6.3 - Proving that a Quadrilateral is a Parallelogram Objectives: To determine whether a quadrilateral is a parallelogram

Theorem 6.5 If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram Since both pairs of opposite sides are congruent, the above quadrilateral is also a parallelogram.

Theorem 6.6 If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Since both pairs of opposite angles are congruent, the above quadrilateral is also a parallelogram.

Theorem 6.7 If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. Since the diagonals are bisected, the above quadrilateral is also a parallelogram.

Theorem 6.8 If one pair of opposite sides of a quadrilateral is both congruent and parallel, then the quadrilateral is a parallelogram. Since one pair of opposite sides are parallel and congruent, the above quadrilateral is also a parallelogram.

Ex: Finding Values for Parallelograms For what values of x and c must MLPN be a parallelogram? P L y 2y - 7 3c - 3 c + 1 3x y + 2 M N

Homework #33 Due Thursday (Dec 06) Page 324 – 325 # 1 – 19 all

Section 6.4 Special Parallelograms Objectives: To use properties of diagonals of rhombuses and rectangles To determine whether a parallelogram is a rhombus or a rectangle

Theorem 6.9 Each diagonal of a rhombus bisects two angles of the rhombus B C 3 4 < 1 congruent < 2 < 3 congruent < 4 1 2 A D

Theorem 6.10 The diagonals of a rhombus are perpendicular C B AC perpendicular BD A D

Theorem 6.11 The diagonals of a rectangle are congruent A D AC congruent BD B C

Theorem 6.12 Theorem 6.13 Theorem 6.14 If one diagonal of a parallelogram bisects two angles of the parallelogram, then the parallelogram is a rhombus. Theorem 6.13 If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus. Theorem 6.14 If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.

Homework # 34 Due Monday (December 10) Page 332 – 334 # 1 – 17 odd # 41 – 46 all

Section 6.5 Trapezoids and Kites Objectives: To verify and use properties of trapezoids and kites

Base Angles -> two angles that share a base of a. trapezoid Base Angles -> two angles that share a base of a trapezoid. Since the bases of the trapezoid are parallel, the two angles that share a leg are supplementary. Base Angles Leg Leg Base Angles

Theorem 6.15 The base angles of an isosceles trapezoid are congruent Ex: Find the measure of each angle in the following trapezoid B C P Q 102° 70° S R A D

Theorem 6.16 The diagonals of an isosceles trapezoid are congruent Theorem 6.17 The diagonals of a kite are perpendicular

Ex: Find the measures of < 1, < 2, and < 3 in the kite B 3 32° 1 2 A C D

Chapter 6 Test Thursday/Friday Homework # 35 Due Monday (December 10) Page 338 – 340 # 1 – 27 odd Quiz Tuesday (Dec 11) Chapter 6 Test Thursday/Friday